Trigonometric derivatives describe how sine, cosine, tangent, and the other trig functions change as an angle changes. They are essential in physics, engineering, signal analysis, waves, circular motion, and any situation involving periodic behavior. On a unit circle, a changing angle creates changing coordinates, and those changes become slopes on the matching sine and cosine graphs.
Learning these derivative patterns makes many calculus problems faster and more meaningful.
Key Facts
- d/dx[sin x] = cos x
- d/dx[cos x] = -sin x
- d/dx[tan x] = sec^2 x
- d/dx[cot x] = -csc^2 x
- d/dx[sec x] = sec x tan x and d/dx[csc x] = -csc x cot x
- Chain rule for trig functions: d/dx[sin u] = cos u · du/dx, where u is a function of x
Vocabulary
- Derivative
- A derivative gives the instantaneous rate of change or slope of a function at a point.
- Unit circle
- The unit circle is a circle of radius 1 centered at the origin, used to connect angles with sine and cosine values.
- Radian
- A radian is an angle measure based on arc length, and trig derivative formulas require angles to be measured in radians.
- Chain rule
- The chain rule is a differentiation rule used when one function is inside another, such as sin(3x) or cos(x^2).
- Periodic function
- A periodic function repeats its values over regular intervals, such as sine and cosine repeating every 2π radians.
Common Mistakes to Avoid
- Using degrees in derivative formulas, which is wrong because d/dx[sin x] = cos x only holds when x is measured in radians.
- Forgetting the negative sign in d/dx[cos x], which is wrong because cosine decreases at x = 0 so its derivative must be -sin x.
- Ignoring the chain rule for expressions like sin(5x), which is wrong because the inside function also changes and contributes a factor of 5.
- Mixing up reciprocal function derivatives, which is wrong because sec x differentiates to sec x tan x, while csc x differentiates to -csc x cot x.
Practice Questions
- 1 Find d/dx[4sin x - 3cos x + 2tan x].
- 2 Find the derivative of y = 7cos(2x) - 5sin(x^2).
- 3 Explain why the derivative of sin x is positive near x = 0 but the derivative of cos x is 0 at x = 0, using the unit circle or graph slopes.